Chaos最新文献

As a regularization of the Hadamard type fractional derivative and a natural extension of the Caputo-Hadamard fractional derivative, the Caputo-Hadamard type fractional derivative exhibits exceptional compatibility, serving as a tractable tool for precise characterization of ultra-slow varying dynamical processes. Compared with Lyapunov stability within the framework of an infinite-time horizon, achieving prescribed performance in finite-time is imperative for practical applications. Herein, this paper concentrates on the finite-time stability of Caputo-Hadamard type fractional differential systems [C-HTFDSs] under two scenarios: systems without delays and systems with proportional delays. To achieve this, for both linear (homogeneous/nonhomogeneous) and nonlinear cases without time delays, the finite-time stability criteria are established leveraging a modified Laplace transform technique and an adaptive fractional Gronwall type inequality, respectively. Then, with regard to the homogeneous and nonhomogeneous linear C-HTFDSs with proportional delays, two novel proportional delayed Mittag-Leffler matrix functions are designed separately, leading to the sound formulations of their fundamental solutions. Finally, as to the nonlinear C-HTFDS with proportional delay, a compatible proportional retarded fractional Gronwall type inequality with two integral terms is constructed and demonstrated in detail. Not only that, several indispensable numerical simulations are implemented to validate the effectiveness and practicality of the theoretical findings.

The concept of transcripts was introduced in 2009 as a means to characterize various aspects of the functional relationship between time series of interacting systems. Based on this concept that utilizes algebraic relations between ordinal patterns derived from time series, estimators for the strength, direction, and complexity of interactions have been introduced. These estimators, however, have not yet found widespread application in studies of interactions between real-world systems. Here, we revisit the concept of transcripts and showcase the usage of transcript-based estimators for a time-series-based investigation of interactions between coupled paradigmatic dynamical systems of varying complexity. At the example of a time-resolved analysis of multichannel and multiday recordings of ongoing human brain dynamics, we demonstrate the potential of the methods to provide novel insights into the intricate spatial-temporal interactions in the human brain underlying different vigilance states.

We present and demonstrate in electronics a chaotic oscillator with an analytic solution that features independent and easily tunable oscillation, symbol, and information time scales. The oscillator is a generalization of a known chaotic oscillator based on an unstable second-order filter that admits an analytic solution. Like most chaotic oscillators, the known oscillator exhibits independently tunable information and oscillation times scales, but directly coupled symbol and oscillation time scales. The generalization presented here separates the symbol time scale from the oscillation time scale in a manner akin to a carrier in a traditional communications system. Importantly, the generalized oscillator also admits an analytic solution. We realize the oscillator at four different pairs of symbol and oscillation time scales in a mixed-signal, electronic circuit that operates at an oscillation frequency of ≈4.2 kHz. We use the theoretical analytic solution to validate the electronic implementation and find good agreement between theoretical system and measured results.

We study an infinite perturbation of the canonical McMillan map using anti-integrability by introducing a perturbation in terms of α. For |α|>1, the McMillan map is no longer integrable. We take the anti-integrable (AI) limit of the map by sending two parameters, α and k^, to infinity. At this limit, the map becomes a non-deterministic relation with three solutions, and the dynamics reduce to a subshift on three symbols. Numerical continuation is applied to periodic AI states to continue onto orbits of the full perturbed map. Results show that certain symbolic sequences are more robust than others in the sense that they continue farther away from the AI limit and closer to the integrable McMillan map: self-symmetric sequences and sequences limited to two of the three symbols.

This study identifies a behavior-driven instability mechanism for spatial pattern formation in ecology, demonstrating that repulsive prey-taxis, a phenomenon where predators avoid well-defended prey, can operate as a primary driver of spatiotemporal complexity through mechanisms distinct from classical diffusion-driven Turing instability. We develop a diffusive predator-prey model incorporating prey-taxis and a Crowley-Martin functional response to capture the interplay between directed movement and predator interference. Our analysis establishes the global existence, uniqueness, and boundedness of classical solutions, ensuring the model's biological well-posedness. We prove that the unique positive equilibrium is globally asymptotically stable under weak attractive prey-taxis, while deriving explicit, ecologically interpretable thresholds for instability. A key finding is that sufficiently strong repulsive prey-taxis (χ<0) induces a novel instability, triggering both Turing and Hopf bifurcations as quantified by the prey-taxis coefficient χ and the conversion rate c. In contrast, attractive prey-taxis (χ>0) exerts a consistent stabilizing effect. Extensive numerical simulations confirm these predictions and unveil a rich spectrum of patterns, ranging from stationary spots, stripes, and labyrinths to dynamic spiral waves and chaos, all of which align with observable ecological phenomena. Our results fundamentally expand the theory of biological pattern formation by establishing prey-taxis, particularly in its repulsive form, as a versatile and potent mechanism for spatial self-organization beyond the effects of pure diffusion.

This work aims to estimate the drift and diffusion functions in stochastic differential equations (SDEs) driven by a special class of Lévy processes with finite jump intensity, using neural networks. We propose a framework that integrates the Tamed-Milstein scheme with neural networks employed as nonparametric function approximators. Estimation is carried out in a nonparametric fashion for the drift function f:R→R and the diffusion coefficient g:R→R. The model of interest is given by dX(t)=f(X(t))dt+g(X(t))dWt+γ∫ZzN(dt,dz), where Wt is a standard Brownian motion and N(dt,dz) is a Poisson random measure on (R+×Z,B(R+)⊗Z,λ(Λ⊗v)), with λ,γ>0, Λ denoting the Lebesgue measure on R+, and v a finite symmetric measure on the measurable space (Z,Z). Neural networks are used as nonparametric function approximators, enabling the modeling of complex nonlinear dynamics without assuming restrictive functional forms. The proposed methodology constitutes a flexible alternative for inference in systems with state-dependent noise and discontinuities driven by Lévy processes.

We consider shear-driven finite-velocity diffusion, both normal and anomalous. In the macroscopic description, this leads to a telegrapher's or Cattaneo-like equation. We analyze the probability density function, and the corresponding moments are obtained analytically. We show that the system exhibits a characteristic crossover of the anomalous dynamics. We also explore corresponding processes under stochastic resetting and find that the systems reach non-equilibrium stationary states in the long time limit that also results in saturation of the evolution of the corresponding mean squared displacement, variance, skewness, and kurtosis.

Symbolic time series analyses are used in economics and other social sciences as a way of reducing the impact of noise on data and to exhibit more clearly the evolution of time series. We show that causality tests applied to symbolic series may fail to detect actual relations or generate statistical artifacts. Well-known causality detection methods, like transfer entropy, Granger's test, or Peter-Clark Momentary Conditional Independence (PCMCI), may miss some existing causal relationships or, more frequently, yield non-existent ones. The performance of these methods may differ, depending on the specific choices of lag structures and alphabet sizes, as well as on the characteristics of the underlying dynamic process.

In the present work, we rely on the helicoidal Peyrard-Bishop model of DNA and use a continuum approximation to solve a crucial dynamical equation of motion. This brings about kink solitary waves moving along the chain. We demonstrate that viscosity is crucial, as no waves are stable when viscosity is neglected. Furthermore, we show that, when viscosity is taken into consideration, the subsonic kink solitons are stable, while the supersonic ones are not.

Climate change is reshaping the global pattern of extremes, driving more frequent heatwaves, synchronous events, and compound hazards. This complexity renders the analysis and prediction of extreme events-such as heatwaves and cold waves-no longer confined to regional or weather-scale processes, as their interactions may trigger cascading and multiscale impacts. Using a complex network framework applied to global reanalysis data from the National Centers for Environmental Prediction and the National Center for Atmospheric Research (1981-2020), we identify robust cross-seasonal teleconnections linking boreal summer heatwaves with winter cold waves, despite their contrasting manifestations. These connections are modulated by large-scale climate modes such as El Niño-Southern Oscillation and the Arctic Oscillation, suggesting that the underlying dynamics transcend seasonal boundaries. This newly revealed relationship exposes previously unexplored correlations, enhances prospects for seasonal-to-interannual prediction of extremes, and underscores the pivotal role of large-scale climate modes in bridging summer and winter variability across hemispheres.